By Deny [1]} (for \(\alpha =2\) , see also H. Cartan [2]}), the cone \(\mathcal {E}^+_\alpha :=\lbrace \mu \in \mathcal {E}_\alpha :\ \mu \geqslant 0\rbrace \)  is strongly complete, and the strong topology on \(\mathcal {E}^+_\alpha \)  is finer than the (induced) vague topology.In B. Fuglede's terminology [3]}, the Riesz kernel is, therefore, perfect. Thus every strong Cauchy net \((\mu _s)\subset \mathcal {E}^+_\alpha \)  converges to the same (unique) limit both strongly and vaguely. This in particular implies that if \(A\subset \mathbb {R}^n\)  is closed (in \(\mathbb {R}^n\) ), then the cone \(\mathcal {E}^+_A:=\mathcal {E}^+_\alpha \cap \mathfrak {M}^+_A\)  is strongly complete, \(\mathfrak {M}^+_A\)  being vaguely closed.
